ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssrabdv Unicode version

Theorem ssrabdv 3146
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1  |-  ( ph  ->  B  C_  A )
ssrabdv.2  |-  ( (
ph  /\  x  e.  B )  ->  ps )
Assertion
Ref Expression
ssrabdv  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2  |-  ( ph  ->  B  C_  A )
2 ssrabdv.2 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ps )
32ralrimiva 2482 . 2  |-  ( ph  ->  A. x  e.  B  ps )
4 ssrab 3145 . 2  |-  ( B 
C_  { x  e.  A  |  ps }  <->  ( B  C_  A  /\  A. x  e.  B  ps ) )
51, 3, 4sylanbrc 413 1  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   A.wral 2393   {crab 2397    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator